where
\({c}\equiv \cos (\frac{\theta }{2}) e^{\frac{-i\zeta }{2}}\)  and
\({s}\equiv \sin (\frac{\theta }{2}) e^{\frac{i\zeta }{2}}\) ,
\(0 \le \theta < \pi \) , \(0 \le \zeta < 2\pi \) , used to write
a generic linear combination of two states. Given the above
presentation, the interpretation of the simplified notations
should be unambiguous. Case (a) has as the initial state a
product of position eigenstate for \(B\)  and a generic state
for \(C\)  (together with \(\left|{\bf 0}\right\rangle _{\!\scriptscriptstyle A}\) ).
The final state maintains being a product state as shown
(involving \(\left|-x\right\rangle _{\!\scriptscriptstyle A}\)  and \(\left|{\bf 0}\right\rangle _{\!\scriptscriptstyle B}\) ).\(C\) , \(A\) , and \(B\)  in the figure (3 of Ref.[1]}) correspond
to \(A\) , \(B\) , and \(C\)  of our notation, respectively. We present (a\(^\prime \) ) as a restriction of the generic state
of \(C\)  to a linear combination of two position eigenstates, for
the purposes of matching to results for a system of qubits.
It captures the key features of the generic case. The
transformation is much like a classical spatial translation
with the `classical' frame of reference for position represented
by a position eigenstate when observed from another frame.
The quantum nature lying in the fact that the position eigenstate
cannot be a momentum eigenstate should be noted. Case
(b\(^\prime \) ) is a restriction of (b) in exactly the same sense.
The cases has the transformation of a product state to one with
nontrivial entanglement (between \(A\)  and \(C\) ), which may be more
easily appreciated from (b\(^\prime \) ). Initial state for case (c) rather
generalized somewhat the one in the figure, as a not necessarily
equal combinations of two perfectly correlated parts of products
of position eigenstates of \(B\) , and \(C\)  with a fixed difference in
eigenvalue. The translation to have \(B\)  as the reference frame
gives the final state as a product with the part for \(C\)  as a simple
eigenstate. The perfect correlation makes all the quantum
fluctuations of \(C\)  unobservable from \(B\) . (d) is really just
a more general form of (c) with the same basic feature. Note that
(b) is much like the inverse of (d). For example, the initial and final
state of (b\(^\prime \) ) with \(c=1\)  and \(s=0\)  can be identified
essentially with the final and initial state of (c) with \(c=1\)  and
\(s=e^{i\zeta ^{\prime }}\) , respectively. We will look at the properties
of the initial and final states in the transformations in (a) and
(d) in the next section.
